use strict;
use Bio::DB::Fasta;

my $db = new Bio::DB::Fasta( "Library.fa" )  # this should be the bigger file
    or die "Failed to create Fasta DB object on Library.fa\n";

open( IDLIST, "Name.fa" );  # this is the list of ID strings
while ( <IDLIST> ) {
    my ($id) = (/^>(\S+)/);  # capture the id string (without the initial ">")
    print ">$id\n", $db->seq( $id ), "\n";
}